Show $\sim:=\{(X,Y)\in\mathcal P(Z)\times\mathcal P(Z): X\cap Y\ne\emptyset\}$ isn't transitive.

Question

Define relation $\sim$ on $\mathcal P(Z)$ as follows: If $X, Y$ are in $\mathcal P(Z)$ (so, subsets of $Z$), we say $X\sim Y$ when $X\cap Y\ne\emptyset$. Show that $\sim$ is not transitive.

The only thing I know about transitive is if $a,b$ belongs to $R$ and $b,c$ belongs to $R$ then $a,c$ must also belong to $R$.

Answer 1

Hint:

For a similar problem, we might be talking about the relation between circles in a plane where one circle is related to another circle if they overlap.

Despite $A$ overlapping $B$ and $B$ overlapping $C$, you can see that $A$ and $C$ do not overlap in the picture below.

Relate this to the specific problem you were asked and formalize a counterexample using the same ideas as above.